Minimal Unsatisfiable Cores Research Articles

Finding Strategyproof Social Choice Functions via SAT Solving

A promising direction in computational social choice is to address research problems using computer-aided proving techniques. In particular with SAT solvers, this approach has been shown to be viable not only for proving classic impossibility theorems such as Arrow's Theorem but also for finding new impossibilities in the context of preference extensions. In this paper, we demonstrate that these computer-aided techniques can also be applied to improve our understanding of strategyproof irresolute social choice functions. These functions, however, requires a more evolved encoding as otherwise the search space rapidly becomes much too large. Our contribution is two-fold: We present an efficient encoding for translating such problems to SAT and leverage this encoding to prove new results about strategyproofness with respect to Kelly's and Fishburn's preference extensions. For example, we show that no Pareto-optimal majoritarian social choice function satisfies Fishburn-strategyproofness. Furthermore, we explain how human-readable proofs of such results can be extracted from minimal unsatisfiable cores of the corresponding SAT formulas.

On getting rid of the preprocessing minimization step in MUC-finding algorithms

When a constraint network is unsatisfiable, it can be of prime importance to provide the network designer with a full-fledged explanation of what causes the absence of any solution to the network. In this respect, minimal unsatisfiable cores (in short, MUCs) form the basis for such an explanation. Efficient MUC extractors are often made of an initial incomplete minimization step that delivers an upper-approximation of a MUC, followed by a refinement step. The first step is believed crucial for the global performance of the whole approach. In this paper, its actual importance is investigated. Especially, it is shown that the first step can be skipped when the refinement process dynamically exploits the information from its own search.

Accelerated Deletion-based Extraction of Minimal Unsatisfiable Cores

Various technologies are based on the capability to find small unsatisfiable cores given an unsatisfiable CNF formula, i.e., a subset of the clauses that are unsatisfiable regardless of the rest of the formula. If that subset is irreducible, it is called a Minimal Unsatisfiable Core (MUC). In many cases, the MUC is required not in terms of clauses, rather in terms of a preknown user-given set of high-level constraints, where each such constraint is a conjunction of clauses. We call the problem of minimizing the participation of such constraints high-level minimal unsatisfiable core (HLMUC) extraction. All the current state-of-the-art tools for MUC- and HLMUC-extraction are deletion-based, which means that they iteratively try to delete clauses from the core. We propose nine optimizations to this general strategy, although not all apply to both MUC and HLMUC. For both cases we achieved over a 2X improvement in run time comparing to the state-of-the-art and a reduction in the core size, when applied to a benchmark set consisting of hundreds of industrial test cases. These techniques are implemented in our award-winning solvers HaifaMUC and HaifaHLMUC.

Boosting MUC extraction in unsatisfiable constraint networks

One very fertile domain of applied Artificial Intelligence is constraint solving technologies. Especially, constraint networks that concern problems that can be represented using discrete variables, together with constraints on allowed instantiation values for these variables. Every solution to a constraint network must satisfy every constraint. When no solution exists, the user might want to know the actual reasons leading to the absence of global solution. In this respect, extracting mucs (Minimal Unsatisfiable Cores) from an unsatisfiable constraint network is a useful process when causes of unsatisfiability must be understood so that the network can be re-engineered and relaxed to become satisfiable. Despite bad worst-case computational complexity results, various muc-finding approaches that appear tractable for many real-life instances have been proposed. Many of them are based on the successive identification of so-called transition constraints. In this respect, we show how local search can be used to possibly extract additional transition constraints at each main iteration step. In the general constraint networks setting, the approach is shown to outperform a technique based on a form of model rotation imported from the sat-related technology and that also exhibits additional transition constraints. Our extensive computational experimentations show that this enhancement also boosts the performance of state-of-the-art DC(WCORE)-like MUC extractors.

Extracting minimal unsatisfiable subformulas in satisfiability modulo theories

Explaining the causes of infeasibility of formulas has practical applications in various fields, such as formal verification and electronic design automation. A minimal unsatisfiable subformula provides a succinct explanation of infeasibility and is valuable for applications. The problem of deriving minimal unsatisfiable cores from Boolean formulas has been addressed rather frequently in recent years. However little attention has been concentrated on extraction of unsatisfiable subformulas in Satisfiability Modulo Theories(SMT). In this paper, we propose a depth-firstsearch algorithm and a breadth-first-search algorithm to compute minimal unsatisfiable cores in SMT, adopting different searching strategy. We report and analyze experimental results obtaining from a very extensive test on SMT-LIB benchmarks.

Improved usability and performance of SMT solvers for debugging specifications

It is now common to construct an extended static checker or software verification system using an SMT theorem prover as the underlying logical verifier. SMT provers have improved significantly in performance over the last several years. However, their usability as a component of software checking and verification systems still has gaps. This paper describes investigations in two areas: the reporting of counterexample information and the testing of vacuity, both of which are important to realistic use of such tools for typical software development. The use of solvers in verification is more effective if the solvers support minimal unsatisfiable cores and incremental construction, evolution and querying of satisfying assignments; current solvers only partially support these capabilities.